Puerto Rico is an unincorporated territory of the United States. As such it cannot accede to international agreements being a non-sovereign nation.
The Department of State administers trade marks, commercial names and US deposits. The latter refers to the deposit of a US Federal Registration with the local Office, similar to the UK registration extension provision in a few Commonwealth countries.
Filings for these rights can be made on-line. It is also possible to make on-line trade mark searches although they state that the database is not yet complete and therefore you should corroborate any findings with the physical Trade Marks Register. It is also necessary to search trade marks, commercial names and US deposits through separate databases.
If you have clicked on the links, you will see that they are in Spanish only. Spanish is the language of an estimated 95% of the population, although English also has official status.
I understand that Puerto Rico does not have design legislation of its own and would be covered automatically by a US design patent granted federally by the USPTO.
Puerto Ricans, despite being US citizens, did not have the right to vote in the Federal election of 6 November 2012. However, on the same day, they voted with regard to their territory's status. They voted in favour of statehood, although like in the US, the island is very divided politcally. If statehood happens then it would see it covered by US Federal registrations (including Madrid Protocol designations) and relegate its Trade Marks Office to the same level as the State trade mark systems of Alabama, Alaska, etc. Statehood would have to be accepted by both the US House and Senate - and it's anticipated it could have a rough journey - so do not expect to need to treat Puerto Rico as a separate trade mark jurisdiction for some time yet.